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Heat oven to 425 degrees with rack in center. Line a rimmed baking sheet with parchment paper. Toss vegetables, pears, and rosemary (2 sprigs) with olive oil and 1/2 teaspoon coarse salt; season with pepper. Spread in a single layer on prepared baking sheet and roast until vegetables are tender and golden, about 30 minutes, stirring once or twice.
While vegetables are roasting, prepare polenta. Combine water, milk, 1 rosemary sprig, and 1 teaspoon salt in a medium saucepan; bring to a boil over medium high; discard rosemary sprig. Slowly add polenta while whisking to prevent lumps. Cook, stirring constantly, until polenta is thick, about 6 to 7 minutes. Remove from heat and whisk in cheese. Divide polenta between four serving bowls and top with roasted vegetables and pears. Garnish with a drizzle of extra-virgin olive oil, ground pepper, and shaved Parmigiano-Reggiano.